Acta Mathematica Academiae Scientiarum Hungaricae 12. (1961)
1961 / 1-2. szám - Foster, F. G.: Queues with batch arrivals. I
QUEUES WITH BATCH ARRIVALS. I By F. G. FOSTER (London) (Presented by A. Rényi) 1. Introduction. The following single server queueing system is considered in this paper: (i) Batches of exactly r units arrive at the sequence of instants, rí, to, .r„, .such that the inter-arrival times, t„+i — t„ >0 (n= 1,2, ...), are identically distributed independent random variables with common distribution function F(x). Put CO CO <jp(s)= \e~s‘dF(x), ct= j xdF(x) and l = \/a. 6 Ó We suppose a < oo. (ii) Units are served individually by a single server. Since the units of a batch arrive simultaneously, we shall suppose that they are ordered for purposes of service. Batches are served in order of arrival. Denote by %n the service time of the n"' unit to be served. We suppose that {%n} (n = 1,2, ...) is a sequence of identically distributed independent positive random variables, independent also of the sequence {t,,}, and that their common distribution function, H(x), is the exponential distribution: H(x) = P[xn^x]== 1— e-r* (x a 0). 00 Put ß=\ xdH(x). Then ,« = 1//?. Define о = Я/(М. ó In the terminology of [3], the system we consider has the 1-input (arrivals) untriggered with input quantity constantly r and a general distribution for the 1-input time. The О-input (departures) is triggered with input quantity constantly unity and an exponential distribution for the О-input time. The system has infinite capacity. On account of the characteristic property of the exponential distribution we have the alternative of supposing that the О-input is untriggered also but with controlled input quantity: the input being virtual whenever the system contains no l’s. In other words, service begins rom time to time whether or not there are any units in the system, and if at 1 Acta Mathematica XII 1—2